JEE Main 2014 · 11 Apr · Shift 1 · Mathematics Q65

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MathematicsSingle correct+4 / -1Question 65 of 90

Let for $i=1,2,3$ , $p_i(x)$ be a polynomial of degree 2 in $x$ , $p_i'(x)$ and $p_i''(x)$ be the first and second order derivatives of $p_i(x)$ respectively. Let,
$A(x)=\begin{bmatrix}p_1(x)&p_1'(x)&p_1''(x)\\p_2(x)&p_2'(x)&p_2''(x)\\p_3(x)&p_3'(x)&p_3''(x)\end{bmatrix}$
and $B(x)=[A(x)]^TA(x)$ . Then determinant of $B(x)$ :

is a polynomial of degree 6 in

x

is a polynomial of degree 3 in

x

is a polynomial of degree 2 in

x

does not depend on

x

Answer and worked solution

Official answer: D

With $p_i=a_ix^2+b_ix+c_i$ , apply the column operation $C_1\to C_1-xC_2+\frac{x^2}2C_3$ . The first column becomes $(c_i)$ and the second $(2a_ix+b_i)$ ; then $C_2\to C_2-xC_3$ leaves $(b_i)$ , with $C_3=(2a_i)$ .

So $\det A$ is constant, and $\det B=(\det A)^2$ does not depend on $x$ .

Topics:Determinants and properties Higher-order derivatives

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JEE Main 2014 · 11 Apr · Shift 1, Mathematics question 65